Post-excision applications dependency map
This map shows the container's direct articles. Nested containers are collapsed and clickable; dependency edges are rolled up from their leaves. fully proved61Proved, and every prerequisite is fully proved too.ready to state1Prerequisites are stated, so this can be written down.planned1Described in the blueprint only.
graph LR
n0["A contractible subspace identifies reduced and relative homology"]:::fully_proved
n1("Excision holds for a union of CW subcomplexes"):::fully_proved
n2["CW-subcomplex regular neighborhoods<br/><small>5 items · 5 fully proved</small>"]:::scope
n3("The double-simplex difference generates reduced homology"):::fully_proved
n4["The ordered difference of the two simplices is a cycle"]:::fully_proved
n5("The first simplex computes homology relative to the second"):::fully_proved
n6["The double simplex is a sphere"]:::fully_proved
n7["The mapping-cone cover has the relative homology of the original pair"]:::fully_proved
n8["The mapping cone of a topological pair"]:::fully_proved
n9["Relative homology is reduced homology of the mapping cone"]:::fully_proved
n10["The pointed wedge is the point quotient of the sigma pair"]:::fully_proved
n11["Relative homology commutes with exact coproducts"]:::fully_proved
n12("The identity simplex generates relative simplex homology"):::fully_proved
n13("A nonempty coproduct of pointed good pairs is good"):::fully_proved
n14["The coproduct pair of a family of pointed spaces"]:::fully_proved
n15["Relative singular chains commute with topological coproducts"]:::fully_proved
n16["Standard-simplex boundary and horn pairs are good"]:::fully_proved
n17["The boundary and zero horn of a standard topological simplex"]:::fully_proved
n18["The standard simplex pair is homeomorphic to the disk pair"]:::fully_proved
n19["The zero face identifies the two simplex point quotients"]:::fully_proved
n20("The zero face induces an isomorphism in relative homology"):::fully_proved
n21["A standard simplex strongly deformation retracts onto its zero horn"]:::fully_proved
n22("The triple connecting map sends a relative cycle to its boundary"):::fully_proved
n23("Reduced homology of a wedge is the direct sum of the summands"):::fully_proved
n24["External chapter: The fundamental group<br/><small>8 items · 8 fully proved</small>"]:::boundary
n25["External chapter: Homology<br/><small>27 items · 25 fully proved · 1 ready to state · 1 planned</small>"]:::boundary
n24 -. 2 .-> n7
n24 --> n8
n24 -.-> n8
n24 -.-> n9
n24 --> n10
n24 -.-> n10
n24 -.-> n23
n25 -->|3| n0
n25 -. 2 .-> n0
n25 --> n1
n25 -. 3 .-> n1
n25 -.-> n3
n25 --> n4
n25 -. 2 .-> n5
n25 --> n7
n25 -. 2 .-> n7
n25 --> n8
n25 -->|2| n9
n25 -.-> n9
n25 --> n10
n25 -. 3 .-> n12
n25 -->|2| n13
n25 --> n14
n25 --> n15
n25 --> n16
n25 -->|2| n17
n25 --> n19
n25 -. 3 .-> n20
n25 --> n21
n25 --> n22
n25 -.-> n22
n25 --> n23
n25 -. 3 .-> n23
n25 --> n2
n0 --> n25
n0 -.-> n3
n0 --> n9
n3 --> n25
n3 --> n6
n4 --> n25
n4 --> n3
n4 --> n5
n4 --> n6
n5 -.-> n3
n6 -->|2| n25
n7 --> n9
n8 --> n7
n8 --> n9
n10 --> n23
n11 --> n25
n11 --> n23
n12 -.-> n3
n13 --> n23
n14 --> n25
n14 --> n10
n14 --> n13
n14 --> n15
n15 -.-> n25
n15 --> n11
n16 -.-> n5
n16 -.-> n20
n17 --> n4
n17 --> n12
n17 --> n16
n17 --> n18
n17 --> n19
n17 --> n20
n17 --> n21
n18 --> n6
n19 -.-> n5
n19 -.-> n20
n20 -.-> n5
n20 -.-> n12
n21 -.-> n12
n22 -.-> n12
n2 -.-> n1
click n0 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#contractible-subspace-relative-reduced-homology" "A contractible subspace identifies reduced and relative homology — 1 item · 1 fully proved"
click n1 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#cw-subcomplex-excision" "Excision holds for a union of CW subcomplexes — 1 item · 1 fully proved"
click n2 "post-excision-applications/cw-subcomplex-neighborhood.html" "CW-subcomplex regular neighborhoods — 5 items · 5 fully proved"
click n3 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-class" "The double-simplex difference generates reduced homology — 1 item · 1 fully proved"
click n4 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-cycle" "The ordered difference of the two simplices is a cycle — 1 item · 1 fully proved"
click n5 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-relative-comparison" "The first simplex computes homology relative to the second — 1 item · 1 fully proved"
click n6 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-sphere-homeomorphism" "The double simplex is a sphere — 1 item · 1 fully proved"
click n7 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-cover-relative-homology" "The mapping-cone cover has the relative homology of the original pair — 1 item · 1 fully proved"
click n8 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-model" "The mapping cone of a topological pair — 1 item · 1 fully proved"
click n9 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-relative-reduced-homology" "Relative homology is reduced homology of the mapping cone — 1 item · 1 fully proved"
click n10 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#pointed-wedge-point-quotient" "The pointed wedge is the point quotient of the sigma pair — 1 item · 1 fully proved"
click n11 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-homology-coproduct" "Relative homology commutes with exact coproducts — 1 item · 1 fully proved"
click n12 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-simplex-fundamental-class" "The identity simplex generates relative simplex homology — 1 item · 1 fully proved"
click n13 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#sigma-good-pair" "A nonempty coproduct of pointed good pairs is good — 1 item · 1 fully proved"
click n14 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#sigma-pointed-pair" "The coproduct pair of a family of pointed spaces — 1 item · 1 fully proved"
click n15 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#singular-relative-chains-coproduct" "Relative singular chains commute with topological coproducts — 1 item · 1 fully proved"
click n16 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-good-pairs" "Standard-simplex boundary and horn pairs are good — 1 item · 1 fully proved"
click n17 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-horn" "The boundary and zero horn of a standard topological simplex — 1 item · 1 fully proved"
click n18 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-disk-pair-homeomorphism" "The standard simplex pair is homeomorphic to the disk pair — 1 item · 1 fully proved"
click n19 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-point-quotient" "The zero face identifies the two simplex point quotients — 1 item · 1 fully proved"
click n20 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-relative-homology" "The zero face induces an isomorphism in relative homology — 1 item · 1 fully proved"
click n21 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-horn-retraction" "A standard simplex strongly deformation retracts onto its zero horn — 1 item · 1 fully proved"
click n22 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#triple-connecting-cycle-formula" "The triple connecting map sends a relative cycle to its boundary — 1 item · 1 fully proved"
click n23 "../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#wedge-reduced-homology" "Reduced homology of a wedge is the direct sum of the summands — 1 item · 1 fully proved"
click n24 "../../../chapters/fundamental-group.html" "The fundamental group — 8 items · 8 fully proved"
click n25 "../../../chapters/homology.html" "Homology — 27 items · 25 fully proved · 1 ready to state · 1 planned"